Preheat the oven to 425 degrees F (220 degrees C). Line a 10x15-inch baking pan with foil.

Brush mushrooms with 1 tablespoon oil. Sprinkle 1/4 teaspoon salt and pepper inside. Put in the prepared pan, stem sides down.

Bake in the preheated oven until tender, about 12 minutes.

Meanwhile, whisk yogurt, mayonnaise, thyme, mustard, lemon juice, and hot sauce in a small bowl to make a smooth sauce.

Heat remaining 1 tablespoon oil in a large skillet over medium heat. Add garlic; cook for 1 minute. Add spinach and Cajun seasoning; cook, stirring, until spinach is just wilted, about 1 minute.

Fill a large saucepan with 2 to 3 inches of water and bring to a boil. Reduce heat to medium-low, add vinegar, and keep water at a gentle simmer. Stir in remaining salt until dissolved. Crack 1 egg into a small bowl, then gently slip egg into simmering water, holding bowl just above the surface. Repeat with remaining eggs. Poach until whites are firm and yolks are thickened but not hard, 3 to 5 minutes. Remove eggs with a slotted spoon.

Microwave sauce until heated through, 30 to 45 seconds. Top mushrooms with spinach mixture, eggs, and warm sauce.